The \(d\) -dimensional euclidean space is denoted by \(\mathbb {R}^d\)  and the space of real matrices of size \(d\)  by \(\mathbb {M}_{d}(\mathbb {R})\) . The space of symmetric matrices of size \(d \times d\)  is denoted by \(\mathbb {S}_d(\mathbb {R})\) . The subset of \(\mathbb {S}_d(\mathbb {R})\)  consisting of positive symmetric matrices is denoted by \(\mathbb {S}^+_d(\mathbb {R})\)  and \(\mathbb {S}^{++}_d(\mathbb {R})\)  is the subset of \(\mathbb {S}_d(\mathbb {R})\)  consisting of definite-positive symmetric matrices. Recall that \(\mathbb {S}^{++}_d(\mathbb {R})\)  is endowed with a smooth (analytic) square root : \(\sqrt{.} : \mathbb {S}^{++}_d(\mathbb {R}) \rightarrow \mathbb {S}^{++}_d(\mathbb {R}) \)  (see for instance [1]} Lemma 5.2.1). Sometimes we will use \(\operatorname{S_p}(A)\)  to denote the set of eigenvalues of a square matrix \(A\) . The euclidean space \(\mathbb {R}^d\)  is endowed with its canonical scalar product : \(x.y := \sum _{i=1}^d x_iy_i \)  and the associated norm \(|x|^2 := \sum _{i=1}^d x_i^2\) . The space \(\mathbb {M}_d(\mathbb {R})\)  is endowed with its canonical scalar product : \(M.N := \operatorname{Tr}(^tMN) \)  and the associated norm \(|M|^2 := \operatorname{Tr}(^tMM)\)  where \(\operatorname{Tr}(M)\)  is the trace of \(M\)  and \(^tM\)  is the transpose of \(M\) . Sometimes we will use another norm on \(\mathbb {M}_d(\mathbb {R})\)  : \(\left. \right.= \sup _{x \in \mathbb {R}^d} \frac{|Mx|}{|x|}\) .
For two real numbers \(r_1\)  and \(r_2\) , \(r_1 \wedge r_2\)  is the minimum of \(r_1\)  and \(r_2\)  and \(r_1 \vee r_2\)  is the maximum of \(r_1\)  and \(r_2\) .
If \(\eta \)  is a \(\sigma \) -finite positive measure on a measurable space \((\Omega , \mathcal {F})\)  and if \(\mu \)  is a \(\sigma \) -finite vector measure on \((\Omega , \mathcal {F})\) , we write \(\mu << \eta \)  if \(\mu \)  is absolutely continuous with respect to \(\eta \)  and we write \(\frac{d\mu }{d\eta } \in L^1(\eta )\)  for the Radon-Nikodym derivative of \(\mu \)  with respect to \(\eta \) .
If \(E\)  is a locally compact, complete, separable metric space and \(l \ge 1\)  is an integer, \(\mathcal {C}_0(E, \mathbb {R}^l) \)  is the space of \(\mathbb {R}^l\) -valued continuous functions on \(X\) , vanishing at infinity. It is endowed with the topology of uniform convergence. Its topological dual \(\left( \mathcal {C}_0(E, \mathbb {R}^l) \right) ^*\)  can be identified thanks to Riesz theorem as the space \(\mathcal {M}(E, \mathbb {R}^l) \)  of \(\mathbb {R}^l\) -valued Borel measures with finite total variation on \(E\) , normed by total variation. We will often consider the weak-* topology on \( \mathcal {M}(E, \mathbb {R}^l) \) . When \(l =1\)  we simply note \(\mathcal {C}_0(E) \)  and \( \mathcal {M}(E)\) . \(\mathcal {M}^+(E) \subset \mathcal {M}(E)\)  is the cone of finite non-negative measures.
The set of Borel probability measures over \(E\)  is denoted by \(\mathcal {P}(E)\) .
If \(r \ge 1\) , \(\mathcal {P}_r(E)\)  is the set of Borel probability measures over \(E\)  with finite moment of order \(r\) . It is endowed with the topology given by the Wasserstein distance \(d_r\)  of order \(r\) . If \(X\)  is a random variable taking values into \((\mathbb {R}^d, \mathcal {B}(\mathbb {R}^d))\) , its law is denoted by \(\mathcal {L}(X) \in \mathcal {P}(\mathbb {R}^d)\) .
We say that \(U : \mathcal {P}_1(\mathbb {R}^d) \rightarrow \mathbb {R}\)  if \(\mathcal {C}^1\)  if there is a continuous function \(\frac{\delta U}{\delta m} :\mathcal {P}_1(\mathbb {R}^d) \times \mathbb {R}^d \rightarrow \mathbb {R}\)  such that, for any \(m_1,m_2 \in \mathcal {P}_1(\mathbb {R}^d)\) ,
\( U(m_1) - U(m_2) = \int _0^1 \int _{\mathbb {R}^d} \frac{\delta U}{\delta m} ( (1-t)m_2 + tm_1,x)(m_1-m_2)(dx)dt. \) 
